Skip to main content

Some New Hilbert's Type Inequalities

Abstract

Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.

1. Introduction

In recent years, several authors [110] have given considerable attention to Hilbert's type inequalities and their various generalizations. In particular, in [1], Pachpatte proved somenew inequalities similar to Hilbert's inequality [11, page 226] involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.

2. Main Results

In [1], Pachpatte established the following inequality involving series of nonnegative terms.

Theorem 2 A.

Let and let and be two nonnegative sequences of real numbers defined for and , where are natural numbers. Let and . Then

(2.1)

where

(2.2)

We first establish the following general form of inequality (2.1).

Theorem 2.1.

Let and . Let and be positive sequences of real numbers defined for and , where are natural numbers. Let and Then

(2.3)

where

(2.4)

Proof.

By using the following inequality (see [12]):

(2.5)

where is a constant, and , , we obtain

(2.6)

Similarly, we have

(2.7)

From (2.6) and (2.7), using Hölder's inequality [13] and the elementary inequality:

(2.8)

where , and we have

(2.9)

Dividing both sides of (2.9) by summing up over from 1 to first, then summing up over from 1 to , using again Hölder's inequality, then interchanging the order of summation, we obtain

(2.10)

This completes the proof.

Remark 2.2.

Taking , (2.3) becomes

(2.11)

Taking and changing , , and into , , and respectively, and with suitable changes, (2.11) reduces to Pachpatte [1, inequality (1)].

In [1], Pachpatte also established the following inequality involving series of nonnegative terms.

Theorem 2 B.

Let , be as defined in Theorem A. Let and be positive sequences for and where are natural numbers. Define and . Let and be real-valued, nonnegative, convex, submultiplicative functions defined on Then

(2.12)

where

(2.13)

Inequality (2.12) can also be generalized to the following general form.

Theorem 2.3.

Let , , and be as defined in Theorem 2.1. Let and be positive sequences for and . Define and Let and be real-valued, nonnegative, convex, submultiplicative functions defined on Then

(2.14)

where

(2.15)

Proof.

By the hypotheses, Jensen's inequality, and Hölder's inequality, we obtain

(2.16)

Similarly,

(2.17)

By (2.16) and (2.17), and using the elementary inequality:

(2.18)

where and we have

(2.19)

Dividing both sides of (2.19) by and summing up over from 1 to first, then summing up over from 1 to , using again inverse Hölder's inequality, and then interchanging the order of summation, we obtain

(2.20)

The proof is complete.

Remark 2.4.

Taking , (2.14) becomes

(2.21)

where

(2.22)

Taking and changing , , and into , , and respectively, and with suitable changes, (2.21) reduces to Pachpatte [1, Inequality ( 7)].

Theorem 2.5.

Let , , , , and , be as defined in Theorem 2.3. Define

(2.23)

for and where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then

(2.24)

Proof.

By the hypotheses, Jensen's inequality, and Hölder's inequality, it is easy to observe that

(2.25)
(2.26)

Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.

Remark 2.6.

In the special case where and , Theorem 2.5 reduces to the following result.

Theorem 2 C.

Let be as defined in Theorem B. Define and for and , where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then

(2.27)

This is the new inequality of Pachpatte in [1,Theorem  4].

Remark 2.7.

Taking and in Theorem 2.5, and in view of , we obtain the following theorem.

Theorem 2 D.

Let , be as defined in Theorem A. Define and for and , where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then

(2.28)

This is the new inequality of Pachpatte in [1,Theorem  3].

References

  1. Pachpatte BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications 1998,226(1):166–179. 10.1006/jmaa.1998.6043

    MATH  MathSciNet  Article  Google Scholar 

  2. Handley GD, Koliha JJ, Pečarić JE: New Hilbert-Pachpatte type integral inequalities. Journal of Mathematical Analysis and Applications 2001,257(1):238–250. 10.1006/jmaa.2000.7350

    MATH  MathSciNet  Article  Google Scholar 

  3. Gao M, Yang B: On the extended Hilbert's inequality. Proceedings of the American Mathematical Society 1998,126(3):751–759. 10.1090/S0002-9939-98-04444-X

    MATH  MathSciNet  Article  Google Scholar 

  4. Jichang K: On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications 1999,235(2):608–614. 10.1006/jmaa.1999.6373

    MATH  MathSciNet  Article  Google Scholar 

  5. Yang B: On new generalizations of Hilbert's inequality. Journal of Mathematical Analysis and Applications 2000,248(1):29–40. 10.1006/jmaa.2000.6860

    MATH  MathSciNet  Article  Google Scholar 

  6. Zhong W, Yang B: On a multiple Hilbert-type integral inequality with the symmetric kernel. Journal of Inequalities and Applications 2007, 2007:-17.

    Google Scholar 

  7. Zhao C-J: Inverses of disperse and continuous Pachpatte's inequalities. Acta Mathematica Sinica 2003,46(6):1111–1116.

    MATH  MathSciNet  Google Scholar 

  8. Zhao C-J: Generalization on two new Hilbert type inequalities. Journal of Mathematics 2000,20(4):413–416.

    MATH  MathSciNet  Google Scholar 

  9. Zhao C-J, Debnath L: Some new inverse type Hilbert integral inequalities. Journal of Mathematical Analysis and Applications 2001,262(1):411–418. 10.1006/jmaa.2001.7595

    MATH  MathSciNet  Article  Google Scholar 

  10. Handley GD, Koliha JJ, Pečarić J: A Hilbert type inequality. Tamkang Journal of Mathematics 2000,31(4):311–315.

    MATH  MathSciNet  Google Scholar 

  11. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.

    Google Scholar 

  12. Németh J: Generalizations of the Hardy-Littlewood inequality. Acta Scientiarum Mathematicarum 1971, 32: 295–299.

    MATH  Google Scholar 

  13. Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 30. Springer, Berlin, Germany; 1961:xii+198.

    Google Scholar 

Download references

Acknowledgments

Research is supported by Zhejiang Provincial Natural Science Foundation of China(Y605065), Foundation of the Education Department of Zhejiang Province of China (20050392). Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and a HKU Seed Grant forBasic Research.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chang-Jian Zhao.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zhao, CJ., Cheung, WS. Some New Hilbert's Type Inequalities. J Inequal Appl 2009, 851360 (2009). https://doi.org/10.1155/2009/851360

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2009/851360

Keywords

  • Real Number
  • Natural Number
  • Convex Function
  • Type Inequality
  • Suitable Modification